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    Chet Aero Marine

    Don’t forget to visit our companion site http://www.vulcanhammer.org

    Use subject to the terms and conditions of the respective websites.


  • ENCE 4610Foundation Analysis and


    Shallow Foundations: Part I

  • Topics for Shallow Foundations, Part I

    ● Types of Shallow Foundations

    ● Spread Footing Design Concept and Procedure

    ● Bearing Capacity Failure Mechanisms Bearing Capacity

    Equation Formulation Bearing Capacity

    Correction Factors

    ● Other Items Local or Punching Shear Factors of Safety Practical Aspects of

    Bearing Capacity Formulations

    Presumptive Bearing Capacities

  • Types of Shallow Foundations● Shallow foundations are usually

    placed within a depth D beneath the ground surface less than the minimum width B of the foundation

    ● Shallow foundations consist of:– Spread and continuous footings– Square, Rectangular or Circular

    Footings– Continuous footings– Ring Foundations– Strap Footings

    ● Wall footings● Mats or Rafts

  • Footings A finite spread footing

    is a shallow foundation that transmits loads and has an aspect ratio of 1 < L/B < 10

    A continuous spread footing is an “infinite” footing where L/B > 10 and the effects of L are ignored

  • Abutment/Wingwall Footing● A situation where the

    shallow foundation is a part of and acts with a retaining wall for both vertical load-bearing and horizontal loads of retained soil

  • Combined Footing● Combined footings are similar to

    isolated spread footings except that they support two or more columns and are rectangular or trapezoidal in shape (Figure 8-7). They are used primarily when the column spacing is non-uniform (Bowles, 1996) or when isolated spread footings become so closely spaced that a combination footing is simpler to form and construct. In the case of bridge abutments, an example of a combined footing is the so-called “spill-through” type abutment (Figure 8-8). This configuration was used during some of the initial construction of the Interstate Highway System on new alignments where spread footings could be founded on competent native soils. Spill-through abutments are also used at stream crossings to make sure that foundations are below the scour depth of the stream.

  • Mat FoundationsMat Foundations A mat is continuous in two

    directions capable of supporting multiple columns, wall or floor loads. It has dimensions from 20 to 80 ft or more for houses and hundreds of feet for large structures such as multi-story hospitals and some warehouses

    Ribbed mats, consisting of stiffening beams placed below a flat slab are useful in unstable soils such as expansive, collapsible or soft materials where differential movements can be significant (exceeding 0.5 inch).

  • Bearing Pressure DistributionBearing Pressure DistributionConcentric LoadsConcentric Loads

    Flexible foundation on clay

    Flexible Foundation on Sand

    Rigid foundation on clay

    Rigid Foundation on Sand

    Simplified Distribution

  • Shear Failure vs. Settlement in Allowable Bearing Capacity

  • Plasticity: Lower and Upper Bound Solutions

    ● The Problem– Bearing Capacity failure is a plastic failure of the soil

    along some failure surface– The problem of plastic failure of twofold:

    ● Finding the failure surface along which the plasticity an ultimately failure takes place

    ● Determining the failure state to which we should design, i.e. lower or upper bound

    ● The first is done by determining which failure surface provides the “path of least resistance” of failure

    ● The second is in part driven by uncertainty requirements in failure

    ● Review of Upper and Lower Bound Concept– Lower Bound: The true failure load is larger than the

    load corresponding to an equilibrium system. The system has failed in at least one place.

    – Upper Bound: The true failure load is smaller than the load corresponding to a mechanism, if that load is determined using the virtual work principle. The system has failed “in general.”

    – The idea is that the true solution is somewhere between the two

    ● We saw this when we went through unsupported cuts in purely cohesive soils– In that case, we had to consider both the shape of

    the failure surface and its location– For slopes, a circular failure surface was

    considered as the most likely failure surface– The actual surface could be located for simple

    slopes using theoretical considerations, but for more complex slopes (layered soils, water table, frictional soils) a trial and error solution was adopted

    ● In principle, only applicable to purely cohesive soils without friction, due to volume expansion considerations– Upper and lower bound theory can be extended to

    soils with a frictional component (or only a frictional component,) but the implementation is much more complicated

    ● We will begin by considering strip (infinite or continuous) foundations only) in cohesive soils

  • Lower Bound Solution

    By direct application and Mohr's Circle

    By theory of elasticity

    The more realistic lower bound

  • Upper Bound Limit Equilibrium MethodUpper Bound Limit Equilibrium Method

    (Circular Failure Surface, Cohesive Soil)(Circular Failure Surface, Cohesive Soil)0

    22 0=

    BBbσBcBbπcBbBBbq=M ultp

    Assume: No soil strength due to internal friction (cohesive soil,) shear strength above foundation base neglectedWe add the effect of the weight of the soil (effective stress) acting on the top of the right

    side of the circle against rotation.







  • More Realistic Upper Bound Case● This is done by moving

    the centre of the failure surface upward

    ● A circular failure surface is what we assumed for slope failure

    ● It is valid for very soft clays, and methods have been developed for use with these types of soils

    ● Soft clays are more subject to settlement

    ● We will not use these in this course

    So the solution is bounded by

  • Development of Prandtl Bearing Capacity Theory

    ● Application of limit equilibrium methods first done by Prandtl on the punching of thick masses of metal (materials with no internal frictional effects)

    ● Prandtl's methods first adapted by Terzaghi to bearing capacity failure of shallow foundations (specifically, he added the effects of frictional materials)

    ● Vesić and others (Meyerhof, Brinch Hansen, etc.) improved on Terzaghi's original theory and added other factors for a more complete analysis

    Note the three zones, the foundation fails along the lower boundary of these zones

  • Assumptions for Bearing Assumptions for Bearing Capacity MethodsCapacity Methods

    Geometric assumption– Depth of foundation is less than

    or equal to its width– Foundation is a strip footing

    (infinite length)*

    Geotechnical Assumptions– Soil beneath foundation is

    homogeneous semi-infinite mass*– Mohr-Coulomb model for soil– General shear failure mode is the

    governing mode*– No soil consolidation occurs– Soil above bottom of foundation

    has no shear strength; is only a surcharge load against the overturning load*

    Foundation-Soil Interface Assumptions Foundation is very rigid

    relative to the soil No sliding occurs

    between foundation and soil (rough foundation)

    Loading Assumptions Applied load is

    compressive and applied vertically to the centroid of the foundation*

    No applied moments present*

    * We will discuss “workarounds” to these assumptions

  • Loads and Failure Zones Loads and Failure Zones on Strip Foundationson Strip Foundations

    Q = load/unit length on foundationq = load/unit area due to effective stress at base of foundationI, II, III = regions of failure in Prantdl theory for bearing capacity failure

  • Basic Equation of Bearing Capacity

  • Basic Equation of Bearing Capacity

    ● Values of Nc, Nq mostly the same. Values of Nγ depend upon theory

    ● DIN/Brinch-Hansen:

    ● CFEM:

    ● Vesić:

    • Vesić-AASHTO Factors:

  • Bearing Capacity Example

  • Types of Bearing Capacity Traditionally, bearing capacity has

    been classified as follows: General Shear (the case

    upon which the theory is based)

    Local Shear Punching Shear Which one takes place

    depends upon consistency or density of soil, which decreases from general to local to punching

    Generally, with softer soils, settlement tends to govern the design more than bearing capacity

  • Soil Property Corrections for Local and Punching Shear

    ● Local and punching shear are accounted for by reducing the cohesion and/or friction angle of the soil

    ● Read entire section on how and when to use this reduction

    ● Also analyze for settlement

  • Bearing Capacity Correction Factors

  • Shape FactorsShape Factors

  • Inclined Base Factors

  • Groundwater Table Correction Factors

  • Embedment Depth Factors

  • Load Inclination Factors A convenient way to account for the

    effects of an inclined load applied to the footing by the column or wall stem is to consider the effects of the axial and shear components of the inclined load individually. If the vertical component is checked against the available bearing capacity and the shear component is checked against the available sliding resistance, the inclusion of load inclination factors in the bearing capacity equation can generally be omitted. The bearing capacity should, however, be evaluated by using effective footing dimensions, as discussed in Section and in the footnote to Table 8-4, since large moments can frequently be transmitted to bridge foundations by the columns or pier walls. The simultaneous application of shape and load inclination factors can result in an overly conservative design.

    Unusual column geometry or loading configurations should be evaluated on a case-by-case basis relative to the foregoing recommendation before the load inclination factors are omitted. An example might be a column that is not aligned normal to the footing bearing surface. In this case, an inclined footing may be considered to offset the effects of the inclined load by providing improved bearing efficiency (see Section Keep in mind that bearing surfaces that are not level may be difficult to construct and inspect. (FHWA NHI-06-089)

  • Allowable Bearing Capacity

  • Bearing Capacity Example Given

    Square shallow foundation, 5’ x 5’

    Foundation depth = 2’ Cohesionless Unit weight 121 pcf Internal friction angle 31

    degrees Load on foundation = 76

    kips Groundwater table very deep

    Find Factor of safety against

    bearing capacity failure Solution

    Governing equation:

    We can neglect factors due to groundwater table (Cw), load inclination (b) and depth (d)

  • Bearing Capacity Example

    Bearing capacity “N” factors for 31 degree friction angle Nc = 32.7 Nq = 20.6 Nγ = 26.0

  • Bearing Capacity Example● Shape Factors● sc = 1+(5/5)(20.6/32.7)

    = 1.63● sγ=1-0.4(5/5) = 0.6● sq = 1+(5/5)(tan(31)) =


  • Bearing Capacity Example Other variables

    c=0 (problem statement) q=(121)(2) = 242 psf γ= 121 pcf (problem

    statement) Substitute and solve

    qult = (0) + (242)(20.6)(1.6) + (0.5)(121)(5)(26.0)(0.6) = 0 + 7976.32 + 4719 = 12,695 psf

    Compute ultimate load Qult = qult * A (12695)(5)(5) = 317,383

    lbs. = 317.3 kips Compute Factor of Safety

    FS = 317.3/76 = 4.17 It’s also possible to do this

    using the pressures qa = 76,000/(25) = 3040

    psf FS = 12,695/3040 = 4.17

  • Effect of Groundwater Table and Layered Soils on Bearing Capacity

    Layered Soils are virtually unavoidable in real geotechnical situations

    Softer layers below the surface can and do significantly affect both the bearing capacity and settlement of foundations

    Pore water pressure increases; reduces both effective stress and shear strength in the soil (same problem as is experienced with unsupported slopes)

    ● Three ways to analyze layered soil profiles:

    Use the lowest of values of shear strength, friction angle and unit weight below the foundation. Simplest but most conservative. Use groundwater factors in conjunction with this.

    Use weighted average of these parameters based on relative thicknesses below the foundation. Best balance of conservatism and computational effort. Use width of foundation B as depth for weighted average

    Consider series of trial surfaces beneath the footing and evaluate the stresses on each surface (similar to slope failure analysis.) Most accurate but calculations are tedious; use only when quality of soil data justify the effort

    ● Groundwater considered using the groundwater correction factor

  • Groundwater Example● Given

    Previous example Groundwater table is 3’ below

    base of foundation● Find

    Bearing Capacity● Solution

    Note that groundwater factor Cwγ is based on Dw, which is distance from surface of soil to groundwater table

    Dw = 2’ + 3’ = 5’ Foundation depth plus

    distance below the base of the foundation

    ● Solution Values for interpolation (from Table 8-

    5) Dw = Df = 2’ (base of foundation):

    Cwq = 1.0 Cwγ = 0.5

    Dw = 1.5 * Bf + Df = (1.5)(5’) + 2’ =9.5’ (bottom of influence zone): = 1.0 Cwq = 1.0 Cwγ = 1.0

    Interpolating: Cwq = 1.0 Cwγ = 0.7

    ● Substitute and solve qult = (0) + (242)(20.6)(1.6)(1.0) + (0.5)

    (121)(5)(26.0)(0.6)(0.7) = 0 + 7976.32 + 3303.3 = 11,280 psf (11% decrease)

  • The Other “Workarounds” for Bearing Capacity Theory

    ● We have already discussed workarounds to the following: Strip footing (shape factors) Level base (base inclination

    factors) Vertical load (load

    inclination factors) Homogeneous soil

    (groundwater factors and related theory)

    General Shear (modified c and φ values)

    ● Other workarounds we will discuss are as follows: Load eccentricity (load is off

    centre or there is a moment that accompanies the load Load eccentricity is

    unavoidable in some circumstances because of the geometry of the structure and the site

    Eccentricity not only impacts bearing capacity but also the basic stability of the foundation base (foundation liftoff)

    Footings on top of or on slopes

  • Eccentric Loading of Foundations● Eccentric loading occurs when

    a footing is subjected to eccentric vertical loads, a combination of vertical loads and moments, or moments induced by shear loads transferred to the footing.

    Abutments and retaining wall footings are examples of footings subjected to this type of loading condition.

    Moments can also be applied to interior column footings due to skewed superstructures, impact loads from vessels or ice, seismic loads, or loading in any sort of continuous frame.

    ● Eccentricity is the distance from the effective point of loading to the centroid of the foundation.

    This distance can be one-way (strip and circular footings) or two-way (square or rectangular footings)

    Eccentricity can occur either because the loading is not at the centroid or there is a moment on the foundation.

    In the case of a moment, the eccentricity is computed by dividing the moment on the foundation by the applied load, i.e., e = M/P

  • Ways of Accounting for Eccentricity

  • Expressing Load Eccentricity and Inclination

    Load Divided by Inclination Angle

    Total Load P Total Vertical Load Pv Total Horizontal Load Ph Angle of Inclination α =

    arctan(Ph/Pv) Load can be concentric or

    eccentric Load and Eccentricity

    Total Vertical Load P Eccentricity from centroid

    of foundation e Horizontal Load (if any)

    not included

    Moment and Eccentric Load

    Total eccentric vertical Load P with eccentricity e

    Replace with concentric vertical load P and eccentric moment M=Pe

    Continuous Foundations Moments, loads

    expressed as per unit length of foundation, thus P/b or M/b

  • Eccentricity and

    Equivalent Footing


  • One Way Loading One-way loading is loading

    along one of the centre axes of the foundation

    Three cases to consider (see right)

    Resultant loads outside the “middle third” result in foundation lift-off and are thus not permitted at all

    After this reduced footing size can be computed

  • Example of One-Way Eccentricity● Given

    Continuous Foundation as shown

    Groundwater table at great depth

    Weight of foundation (concrete) not included in load shown

    ● Find Whether resultant force

    acts in middle third Minimum and maximum

    bearing pressures Reduced bearing area

  • Example of One Way Example of One Way EccentricityEccentricity

    Compute Weight of Foundation Wf/b = (5)(1.5)(150) = 1125 lb/ft

    Compute eccentricity

    Thus, eccentricity is within the “middle third” of the foundation and foundation can be analysed further without enlargement at this point

    e=(M /b )Q /b

    = 800012000+1125

    =0 .61 ft .


    6=0 .833 ft .>0 .61 ft .

  • Equations for One-Way Pressures Equations for One-Way Pressures with Eccentric/Moment Loadswith Eccentric/Moment Loads

    If q at any point is less than zero, resultant is outside the middle third

    Wf is foundation weight

  • Example of One Way Example of One Way EccentricityEccentricity

    Compute minimum and maximum bearing pressures

  • Example of One-Way Eccentricity● Since the resultant is

    within the middle third, we can compute the reduced foundation size B’

    ● As this is a continuous footing experiencing one-way eccentricity, we do not need to consider an L’

    ● From previous computations: B = 5’ e = 0.61’

    ● Reduced Foundation Width B’ = 5 – (2)(0.61) = 3.78’

  • Two-Way Eccentricity• Eccentricity in both “B” and “L”

    directions produces a planar distribution of stress

    • Kern of Stability Foundation stable

    against overturn only if resultant falls in the kern in the centre of the foundation

    Resultant in the kern if6 eBB

    +6 eLL


    eB, e

    L = eccentricity in B, L directions

  • Bearing Pressure at CornersBearing Pressure at CornersTwo-Way EccentricityTwo-Way Eccentricity

    • Helpful hint to prevent confusion of eccentricity of finite vs. infinite (continuous) foundationso Always use one-way

    eccentricity equations for continuous foundations

    o Always use two-way eccentricity equations for finite foundations

    o Two-way equations will reduce to one-way equations if one of the eccentricities (eB, eL) is zero






    WQq LBf


  • Two-Way Eccentricity Example Given

    Grain silo design as shown Each silo has an empty

    weight of 29 MN; can hold up to 110 MN of grain

    Weight of mat = 60 MN Silos can be loaded

    independently of each other Find

    Whether or not eccentricity will be met with the various loading conditions possible

    Eccentricity can be one-way or two-way

  • Two-Way Eccentricity Two-Way Eccentricity ExampleExample

    One-Way Eccentricity Largest Loading: two

    adjacent silos full and the rest empty

    Q = (4)(29) + 2(110) + 60 = 396 MN

    M = (2)(110)(12) = 2640 MN-m

    e= MQ

    e= 2640396

    e=6 .67mB6


    =8 .33m>6. 67m

    Eccentricity OK for one-way eccentricity

  • Two-Way Eccentricity Two-Way Eccentricity ExampleExample

    Two-Way Eccentricity Largest Loading: one silo full and the rest

    empty P = (4)(29) + 110 + 60 = 286 MN MB = ML = (110)(12) = 1320 MN-m


    = 1320286

    =4 .62m

    6 eBB


    =2( ( 6 ) ( 4 .62 )50 )=1 .11>1Not acceptable

  • Two-Way Eccentricity Two-Way Eccentricity ExampleExample

    Two-Way Eccentricity Solution to Eccentricity

    Problem: increase the size of the mat

    Necessary to also take other considerations into account (bearing failure, settlement, etc.)

    6 eBB


    =2( ( 6 ) ( 4 .62 )B )=1B=L=55. 4m

  • Equivalent Footing Using Two-Equivalent Footing Using Two-Way Eccentricity ExampleWay Eccentricity Example

    Largest Loading: one silo full and the rest empty

    Result of Two-Way Eccentricity Analysis

    eB = eL = 4.62 m B = L = 55.4 m (expanded

    foundation) Equivalent Footing

    Dimensions B’ = B – 2eB = 55.4 – (2)(4.62) B’ = 45.8 m = L’ (as B = L and

    eB = eL)

  • Equivalent Footing Using Two-Equivalent Footing Using Two-Way Eccentricity ExampleWay Eccentricity Example

    One-Way Eccentricity Largest Loading: two

    adjacent silos full and the rest empty

    B = L = 55.4 m (expanded foundation)

    eB = 6.67m eL = 0 m B’ = B’-2eB = 55.4 – (2)

    (6.67) = 42.1 m L = L’ = 55.4 m B


    6=8 .33m>6. 67m

  • Other Notes on Bearing Other Notes on Bearing Capacity FactorsCapacity Factors

    • Two ways to handle B’ and L’ values when computing shape factors (which are a function of B/L):o AASHTO (2002) guidelines recommend calculating the shape factors, s, by using the effective

    footing dimensions, B and L’.′ and L’.o However, the original references (e.g., Vesi , 1975) do not specifically recommend using the ć, 1975) do not specifically recommend using the

    effective dimensions to calculate the shape factors. Since the geotechnical engineer typically does not have knowledge of the loads causing eccentricity, use full footing dimensions be used to calculate the shape factors for use in computation of ultimate bearing capacity.

    o Either is acceptable for problems in this class. In practice, which one you would use would depend upon a) the project (a highway project would tend to use AASHTO recommendations) and how well the location of the loads was known.

    • Bowles (1996) also recommends that the shape and load inclination factors (s and i) should not be combined.

    • In certain loading configurations, the designer should be careful in using inclination factors together with shape factors that have been adjusted for eccentricity (Perloff and Baron, 1976). The effect of the inclined loads may already be reflected in the computation of the eccentricity. Thus an overly conservative design may result.

  • Bearing Capacity for Foundation at Top of a SlopeBearing Capacity for Foundation at Top of a Slope

  • Example of Footings Example of Footings on Slopeson Slopes

    • Solutiono Ncq for D/B = 0 and Slope Angle

    of 30 deg. = 4.9

    o Ncq for D/B = 1 and Slope Angle of 30 deg. = 6.4

    o Linearly interpolating, Ncq =

    (6.4+4.9)/2 = 5.7

    o Nγq = 1 since the soil is purely cohesive

    o B/2 = Do Shape factors are unity because it

    is a continuous footing; water table and embedment factors are like wise not considered

    – qult = (75)(5.7)+(1.5)(18.5) =

    455.25 kPa

    • Giveno Strip footing to be constructed on top

    of the slopeo Soil properties: c = 75 kPa, γ = 18.5

    kN/m3, water table very deepo H = 8 m, B = 3 m, D = 1.5 m, b = 2

    m, Slope Angle = 30 deg.

    • Findo Ultimate Bearing Capacity of

    Footing, using solution method of previous slide

    o Solutiono B < H since 3 m < 8 mo Obtain Ncq from Figure 8-18(e) for

    Case I with No = 0o D/B = 1.5/3 = 0.5o b/B = 2/3 = 0.667

  • Required Footing Required Footing SetbacksSetbacks

  • Practical Aspects of Bearing Capacity Formulations

  • Failure Zones for Bearing Capacity

  • Presumptive Bearing CapacityPresumptive Bearing Capacity

  • Presumptive Bearing Capacity on Rock

  • Questions?

    ENCE 4610 Foundation Analysis and DesignTopics for Shallow Foundations, Part ITypes of Shallow FoundationsFootingsAbutment/Wingwall FootingCombined FootingSlide 7Slide 8Shear Failure vs. Settlement in Allowable Bearing CapacityPlasticity: Lower and Upper Bound SolutionsLower Bound SolutionSlide 12More Realistic Upper Bound CaseDevelopment of Prandtl Bearing Capacity TheorySlide 15Slide 16Basic Equation of Bearing CapacitySlide 18Bearing Capacity ExampleTypes of Bearing CapacitySoil Property Corrections for Local and Punching ShearBearing Capacity Correction FactorsSlide 23Inclined Base FactorsGroundwater Table Correction FactorsEmbedment Depth FactorsLoad Inclination FactorsAllowable Bearing CapacitySlide 29Slide 30Slide 31Slide 32Effect of Groundwater Table and Layered Soils on Bearing CapacityGroundwater ExampleThe Other “Workarounds” for Bearing Capacity TheoryEccentric Loading of FoundationsWays of Accounting for EccentricityExpressing Load Eccentricity and InclinationSlide 39One Way LoadingExample of One-Way EccentricitySlide 42Slide 43Slide 44Slide 45Two-Way EccentricitySlide 47Two-Way Eccentricity ExampleSlide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Practical Aspects of Bearing Capacity FormulationsFailure Zones for Bearing CapacitySlide 60Presumptive Bearing Capacity on RockSlide 62